Finite Group Databasebeta

← C31×#11551 #21468310

Subgroups — C31×#11551 #21468310

Order 7936.

398 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2503C211yesmaximal, characteristic
2C31×#1463C211yesmaximal, characteristic
3C2×C31×#324C211yesmaximal, characteristic
4C31×#1324C212yesmaximal, complemented
5C31×#1319C212yesmaximal, complemented
6C2×C31×C22⋊Q8C411yescharacteristic
7C22×C4⋊C4×C31C2211yescharacteristic
8C31×#407—21no—
9C2×C31×C22⋊C8—21no—
10C31×#400—21no—
11C31×#400—21no—
12C31×#391C411yescharacteristic
13C31×#391C2211yescharacteristic
14C31×#382C411yescharacteristic
15C31×#382C2211yescharacteristic
16C31×#381—21no—
17C31×#381—21no—
18C4×C22⋊C4×C31C411yescharacteristic
19C31×#324—21no—
20C31×#324—21no—
21C31×#324C2212yes—
22C31×#324C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C23×C4×C31C2×C411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—41no—
28C22×C8×C31—41no—
29C31×C22⋊Q8—41no—
30C31×C22⋊Q8—41no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—41no—
33C2×C4⋊C4×C31—41no—
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C4⋊C4×C31C2311yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31C2×C411yescharacteristic
42C2×C42×C31C2×C411yescharacteristic
43C31×C22⋊C8—22no—
44C31×C22⋊C8—22no—
45C31×C22⋊C8—41no—
46C31×C22⋊C8—41no—
47C31×#95—21no—
48C31×#95—21no—
49C31×#95—21no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C24×C31C22⋊C411yescharacteristic
57C2×Q8×C31—22no—
58C2×Q8×C31—41no—
59C2×Q8×C31—41no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—22no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—41no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22×C411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80M16×C31—82no—
81C2×C8×C31—42no—
82C2×C8×C31—42no—
83C2×C8×C31—82no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—41no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—41no—
93C42×C31—22no—
94C42×C31—41no—
95#11551C3111yesmaximal, characteristic, direct product factor, Sylow
96C23×C31—22no—
97C23×C31—21no—
98C23×C31—21no—
99C23×C31—41no—
100C23×C31#9911yescharacteristic
101C23×C31#9911yescharacteristic
102C23×C31C2×C22⋊C411yescharacteristic
103Q8×C31—82no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—81no—
133C2×C4×C31—81no—
134C2×C4×C31#13112yes—
135C8×C31—82no—
136C8×C31—82no—
137#2503C2×C3111yescharacteristic
138#1463C2×C3111yescharacteristic
139C2×#324C2×C3111yescharacteristic
140#1324C2×C3112yescomplemented
141#1319C2×C3112yescomplemented
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—42no—
150C22×C31—42no—
151C22×C31#35412yes—
152C22×C31#40811yescharacteristic
153C22×C31C2×#9911yescharacteristic
154C22×C31#41311yescharacteristic
155C4×C31—22no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—81no—
162C4×C31—81no—
163C4×C31—81no—
164C4×C31—81no—
165C2×C22⋊Q8C4×C3111yescharacteristic
166C22×C4⋊C4C22×C3111yescharacteristic
167#407—21no—
168C2×C22⋊C8—21no—
169#400—21no—
170#400—21no—
171#391C4×C3111yescharacteristic
172#391C22×C3111yescharacteristic
173#382C4×C3111yescharacteristic
174#382C22×C3111yescharacteristic
175#381—21no—
176#381—21no—
177C4×C22⋊C4C4×C3111yescharacteristic
178#324—21no—
179#324—21no—
180#324C22×C3112yes—
181#324C22×C3112yes—
182C2×C31—21no—
183C2×C31—21no—
184C2×C31—22no—
185C2×C31—41no—
186C2×C31#146811yescharacteristic
187C2×C31C2×#35411yescharacteristic
188C2×C31#213311yescharacteristic
189C22×Q8C2×C4×C3111yescharacteristic
190C23×C4C2×C4×C3111yescharacteristic
191C23×C4D4×C3111yescharacteristic
192C23×C4D4×C3111yescharacteristic
193C2×M16—41no—
194C22×C8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×C4×C3111yescharacteristic
201C2×C4⋊C4C2×C4×C3111yescharacteristic
202C2×C4⋊C4D4×C3111yescharacteristic
203C2×C4⋊C4D4×C3111yescharacteristic
204C2×C4⋊C4C23×C3111yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42C2×C4×C3111yescharacteristic
208C2×C42C2×C4×C3111yescharacteristic
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222C31#1155111yescharacteristic, direct product factor, Sylow
223C24C22⋊C4×C3111yescharacteristic
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C22⋊C4×C3111yescharacteristic
242C22×C4C22⋊C4×C3111yescharacteristic
243C22×C4C22⋊C4×C3111yescharacteristic
244C22×C4C22×C4×C3111yescharacteristic
245C22×C4C2×D4×C3111yescharacteristic
246C22×C4C2×D4×C3111yescharacteristic
247M16—82no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—82no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—22no—
261C42—41no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—41no—
266C23C31×#9911yescharacteristic
267C23C31×#9911yescharacteristic
268C23C2×C22⋊C4×C3111yescharacteristic
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4C31×#13112yes—
301C8—82no—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22C31×#35412yes—
313C22C31×#40811yescharacteristic
314C22C2×C31×#9911yescharacteristic
315C22C31×#41311yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2C31×#146811yescharacteristic
331C2C2×C31×#35411yescharacteristic
332C2C31×#213311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C22×C4⋊C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C31×#1324C2semidirect
5. C31×#1319C2semidirect
95. #11551C31direct (N × H)
140. #1324C2×C31semidirect
141. #1319C2×C31semidirect
222. C31#11551direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#25032
C31×#14632
C2×C31×#3242
C31×#13242
C31×#13192
#1155131